Lecture Notes

$\gdef\End{\text{End}}$ Let $M=\R^2$ and $L$ be the trivial line bundle on $M$. We identify sections of $L$ with smooth function on $M$. Let $\nabla = d + A$, where $d$ is the trivial connection on $L$ and $A$ is the connection 1-form in $\Omega^1(M, \End(L)) = \Omega^1(M)$: $A = x dy - y d x$ Let point $a=(1,0)$, $b=(-1,0)$, and $\gamma_\pm$ be path from $a$ to $b$, going along upper (or lower) semicircle: $\gamma_\pm: [0,\pi] \to \R^2, \quad t \mapsto (\cos t, \pm \sin t).$

Question: compute the parallel transport along $\gamma_+$ and $\gamma_-$.

Let $M = \R^2$, and $E$ the trivial rank-2 vector bundle on $\R^2$. Let $\nabla = d + A,$ where $d$ is the trivial connection on $L$ and $A$ is the connection 1-form in $\Omega^1(M, \End(L)) = \Omega^1(M) \otimes M_2(\R)$: $A = \begin{pmatrix} 1 & 0 \cr 0 & -1 \end{pmatrix} dx + \begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix} dy$ Compute the curvature 2-form $F \in \Omega^2(M)\otimes M_2(\R)$.

Optional: Compute the parallel transport along the boundary of the unit square $[0,1]^2$, starting from $(0,0)$ in counter-clockwise fashion. (Hint: you will end up with answer like $\beta^{-1}\alpha^{-1}\beta\alpha$ where $\alpha$ and $\beta$ are in $GL(2, \R)$.